Problem: Solve the exponential equation for $x$. 49 4 x − 1 7 x − 6 = 7 8 x + 3 \dfrac{49\^{ 4x-1}}{7\^{ x-6}}=7\^{ 8x+3} $x=$
Solution: The strategy Let's write $49$ in base $7$. Then, using the properties of exponents, we can express the entire left hand side of the equation as $7$ raised to some linear function. Finally, we can equate the exponents of the resulting equation to solve for the unknown. Simplifying the left hand side 49 4 x − 1 7 x − 6 = ( 7 2 ) 4 x − 1 7 x − 6 = 7 8 x − 2 7 x − 6 = 7 8 x − 2 − ( x − 6 ) = 7 7 x + 4 ( 49 = 7 2 ) ( ( a n ) m = a n ⋅ m ) ( a n a m = a n − m ) \begin{aligned}\dfrac{49\^{ 4x-1}}{7\^{ x-6}}&=\dfrac{(7^2)\^{ 4x-1}}{7\^{ x-6}}&&&&(49=7^2) \\\\\\\\ &=\dfrac{7\^{ C{8x-2}}}{7\^{ {x-6}}} &&&&((a^n)^m=a^{n\cdot m})\\\\\\\\ &=7\^{ C{8x-2} \ - \ ({x-6})}&&&&(\dfrac{a^n}{a^m}=a^{n-m})\\\\\\\\ &=7\^{7x+4} \end{aligned} Solving the equation We obtain the following equation. 7 7 x + 4 = 7 8 x + 3 7\^{ 7x+4}=7\^{ 8x+3} Now we can equate the exponents and solve for $x$. $\begin{aligned} 7x+4 &=8x+3\\\\ x &=1\end{aligned}$ The answer The answer is $x=1$. You can check this answer by substituting $\it{x=1}$ in the original equation and evaluating both sides.